One job of sensor technology is the contact-free, precise measurement of distances and speeds. Microwaves, light waves or ultrasound are utilized for this purpose. Such sensors are versatilely employed, for example in industrial automation, in automotive technology and in the household. Radar sensors according to the FMCW principle (frequency modulated continuous wave) are standard.
FIG. 1 shows a typical circuit diagram of a FMCW sensor (also see A. G. Stove, xe2x80x9cLinear FMCW Radar Techniques.xe2x80x9d, IEE Proc. F 139, 343-350 (1992)). The signal source is an oscillator no that can be frequency-modulated. This oscillator is detuned time-dependently in frequency via a drive unit. The sensor emits the transmission signal s(t) via the transmission/reception unit SEE, and receives a reception signal r(t) delayed in time corresponding to the running time to the target. A separation of transmission and reception signals is effected, for example, by a transmission and reception diplexer SEW; for instance, (a circulator or a directional coupler. Alternatively, separate antennas can be employed for transmission and reception (what is referred to as a bistatic arrangement). The measured signal mess(t) generated in a first mixer MI1, which corresponds to the mixed product (difference frequency) of transmission signal s(t) and reception signal r(t), is filtered with a low-pass filter TP1. For generating a measured signal that allows a presentation as a complex number, the reception signal r(t) can be mixed in a second mixer M12 with the transmission signal shifted in phase by xcfx80/2 in a phase shifter PH and can be subsequently filtered in a second low-pass filter TP2. The evaluation in an evaluation unit AE is preferably constructed with a digital signal processor to which the measured signal digitalized with analog-to-digital converters A/D is supplied as real part cosine and imaginary part sine of the complex signal. The frequency modulation of the transmission signal usually occurs linearly in time (see H. D. Griffiths, xe2x80x9cThe Effect of Phase and Amplitude Errors in FM Radarxe2x80x9d, IEEE Colloquium on High Time-Bandwidth Product Waveforms in Radar and Sonar, London, UK, May 1, 1991, pages 9/1-9/5).
DE 195 33 124 discloses a method wherein errors in the frequency modulation are detected and corrected upon employment of a delay line. Such a delay line is shown in drawing FIG. 1, this comprising a delay element T (preferably a surface wave component) and a further mixer RMI. Preferably, a further low-pass filter TP3 follows. In the further mixer RMI, the delayed signal is mixed with the current signal to form a reference signal. A correction signal is calculated in the evaluation unit AE, this correction signal then also potentially serving the purpose of undertaking a correction of the frequency modulation via a programmable drive unit. The drawing FIG. 1 shows a sensor system wherein the transmission frequency before the delay is mixed onto a lower intermediate frequency in a mixer ZFMI. The local oscillator LO is provided for this purpose, this supplying a lower frequency than the signal source MO.
In any case, the sensor signal mess(t) of a FMCW radar sensor is composed of a superimposition of discrete sine oscillations whose frequencies represent the quantities to be measured (distance, running time, velocity). Given, for example, a filling level sensor, the frequency of the sensor signal is proportional to the difference between container height and filling height; given the presence of a plurality of targets, correspondingly more frequencies occur. When a target moves, than the distance-dependent frequency of the sensor signal has an additional Doppler frequency additively superimposed on it. In the specific case of a CW radar sensor, only this Doppler frequency is detected, this representing the target velocity.
A Fourier transformation is usually employed for interpreting the frequency spectrum of measured signals. In this xe2x80x9cFourier analysisxe2x80x9d, the resolution with which neighboring frequencies can be separated is limited due to various influences. Given FMCW sensors, the limiting resolution also derives from the bandwidth of the frequency modulation limited by the technology or by approval stipulations. On the other hand, an optimally high frequency resolution is desirable so that noise frequencies can be separated from the signal frequencies in as far-reaching way as possible.
The application of auto-regressive methods (AR; specific parametric modeling methods) has been discussed in the literature for enhancing the frequency resolution of FMCW sensors (L. G. Cuthbert et al, xe2x80x9cSignal Processing In An FMCW Radar For Detecting Voids and Hidden Objects In Building Materialsxe2x80x9d, in I. T. Young et al (editors), Signal Processing III: Theories and Applications, Elsevier Science Publishers B. V., EURASIP 1986). In fact, a higher frequency resolution can be achieved with modern parametric modeling methods (AR, MA and ARMA; S. M. Kay, xe2x80x9cModem Spectral Estimation, Theory and Applicationxe2x80x9d, Prentice Hall, Englewood cliffs, N.J., 1988) than with Fourier analysis, but only with an adequately great signal-to-noise ratio of the measured signal, which must amount to more than 40 dB given a typical data length of N=100. This prerequisite, however, is generally not met in a FMCW sensor.
Methods for analysis of frequency spectra wherein mathematical investigations deriving from Prony are combined with what is referred to as a singular value decomposition (referred to below as SVD) are described in the text book by S. L. Marple, Jr., xe2x80x9cDigital Spectral Analysis With Applicationsxe2x80x9d, Prentice Hall, Englewood Cliffs, N.J., 1988. These methods are always especially successful when the signal x(n) is composed of a finite number p of discrete frequencies fk (with the amplitudes ck) and noise R(n) according to                                           x            ⁡                          (              n              )                                =                                                    ∑                                  k                  =                  1                                p                            ⁢                                                c                  k                                ⁢                                  ⅇ                                      2                    ⁢                                          xe2x80x83                                        ⁢                    π                    ⁢                                          xe2x80x83                                        ⁢                    ⅈ                    ⁢                                          xe2x80x83                                        ⁢                                          f                      k                      n                                                                                            +                          R              ⁡                              (                n                )                                                    ,                  xe2x80x83                ⁢                  n          =          1                ,        2        ,        3        ,        …        ⁢                  xe2x80x83                ,        N        ,                            (        1        )            
This is the case in FMCW sensors, particularly in combination with the above-described distortion-correction with a surface wave delay line.
The frequencies fk and their plurality p as well as the corresponding (complex) amplitudes ck are to be determined from the finite data set [x(1), x(2), . . . , x(N)]. In the underlying Prony method (without SVD), what are referred to as the FBLP coefficients (forward backward linear prediction) a (k) (k=0, . . . , p) are calculated from x(n) in a first step. These satisfy the equations
(0)x(n)+a(1)x(nxe2x88x921)+ . . . +a(p)x(nxe2x88x92p)=0xe2x80x83xe2x80x83(2
and, with the complex, conjugated data x*(n) in reverse sequence
a(0)x*(nxe2x88x92p)+a(1)x*(nxe2x88x92p+1)+ . . . +a(p)x*(n)=0xe2x80x83xe2x80x83(2b)
for n=p+1, p+2, . . . , N.
A total of N-p of such linear equation pairs (2a, 2b) can be erected with the given data set [x(1),x(2), . . . , x(N)], i.e. 2(Nxe2x88x92p) equations; the a(k) can be determined therefrom. In general (when N greater than 2p applies), the equation system (2a, 2b) is composed of more equations than of unknowns a(k). The method according to the principle of the least square (referred to below as LS) is then utilized for a solution, and, as a result thereof, averaged via noise influences due to the noise components R(n) from equation (1): for an over-defined, linear equation system A * x=b, the LS solution is generally established by
x=(AH*A)xe2x88x921*AH*b
(see, for example, the textbook of Marple, Page 77).
The sought signal frequencies fk are then calculated in a second step from the zero settings zk of the polynomial P9z) formed with the a(k) according to                               P          ⁡                      (            z            )                          =                              ∑                          k              =              0                        p                    ⁢                                    a              ⁡                              (                k                )                                      ⁢                          z                              p                -                k                                                                        (        3        )            
When, for instance, zk is one of the p zero settings, i.e. P(Zk)=0 applies, then
exp(2xcfx80xc2x7ixc2x7fk)=zk.xe2x80x83xe2x80x83(4)
When the frequencies are identified, then the equations                                           x            ⁡                          (              n              )                                =                                    ∑                              k                =                1                            p                        ⁢                                          c                k                            ⁢                              ⅇ                                  2                  ⁢                                      xe2x80x83                                    ⁢                  π                  ⁢                                      xe2x80x83                                    ⁢                  ⅈ                                            ⁢                              f                k                n                                                    ,                  xe2x80x83                ⁢                  n          =          1                ,        2        ,        3        ,        …        ⁢                  xe2x80x83                ,        N        ,                            (        5        )            
generally form an over-defined, linear equation system for the p amplitudes ck that is solved in a third step by LS methods (see above), as a result whereof averaging is again carried out over noise influences by the R(n) components.
The results that can be achieved with this method are more beneficial than with Fourier analysis only given a high signal-to-noise ratio of the measured signal. Moreover, the model parameter p is generally not known and the identification thereof is problematical. Improvements are achieved by application of SVD (D. W. Tufts, R. Kumaresan , xe2x80x9cEstimation of Frequencies of Multiple Sinusoids: Making Linear Prediction Perform Like Maximum Likelihoodxe2x80x9d, Proc. IEEE 70, 979-989 (1982)).
Far more FBLP coefficients a(k) are introduced than the plurality p of anticipated frequencies, for instance L( greater than p). The equations analogous to (2a) and (2b) then read
a(0)x(n)+a(1)x(nxe2x88x921)+ . . . +a(L)x(nxe2x88x92L)=0,xe2x80x83xe2x80x83(6a)
a(0)x*(nxe2x88x92L+a(1)x*(nxe2x88x92L+1)+ . . . +a(L)x*(n)=0,xe2x80x83xe2x80x83(6b)
for n=L+1, L+2, . . . , N with p less than L less than Nxe2x88x92p.
These Nxe2x88x92L pairs of equations can be combined to
[a(L), a(Lxe2x88x921), . . . , a(0)]xc2x7DNL=0xe2x80x83xe2x80x83(7)
(xc2x7 references the matrix multiplication) with       D    NL    =      "AutoLeftMatch"          [                                                  x              ⁡                              (                1                )                                                                        x              ⁡                              (                2                )                                                          …                                              x              ⁡                              (                                  N                  -                  L                                )                                                                        x              *                              (                N                )                                                                        x              *                              (                                  N                  -                  1                                )                                                          …                                              x              *                              (                                  L                  +                  1                                )                                                                                        x              ⁡                              (                2                )                                                                        x              ⁡                              (                3                )                                                          …                                              x              ⁡                              (                                  N                  -                  L                  +                  1                                )                                                                        x              *                              (                                  N                  -                  1                                )                                                                        x              *                              (                                  N                  -                  2                                )                                                          …                                              x              *                              (                L                )                                                                          ⋮                                ⋮                                ⋮                                ⋮                                ⋮                                ⋮                                ⋮                                ⋮                                                              x              ⁡                              (                                  L                  +                  1                                )                                                                        x              ⁡                              (                                  L                  +                  2                                )                                                          …                                              x              ⁡                              (                N                )                                                                        x              *                              (                                  N                  -                  L                                )                                                                        x              *                              (                                  N                  -                  L                  -                  1                                )                                                          …                                              x              *                              (                1                )                                                        ]      
Here, the measured data are arranged in a matrix, what is referred to as a double Hankel matrix. Each column of this matrix, multiplied by the line vector [a(l), a(Lxe2x88x921), . . . , a(0)], supplies one of the equations (6a, 6b). If the signal x(n) were to contain no noise, i.e. only p exponential oscillations, then the equation system for determining the L coefficients a(k) would be over-defined, i.e. only p of the equations (6a) and (6b) would be linearly independent. Accordingly, the matrix DNL would have the rank p( less than L). When, however, noise is present, then DNL generally has the maximum rank (2Nxe2x88x922L or L+1) and only approximately has the lower rank p.
With the assistance of what is referred to as a xe2x80x9csingular value decompositionxe2x80x9d(SVD), a matrix is defined from DNL that has the rank p and is an approximation of DNL in the sense of the LS method (least sum of the square of the deviations) known from the literature. According to
DNL=U*S*VHxe2x80x83xe2x80x83(8)
the SVD resolves the matrix DNL into two unitary matrices U and V (VH is the adjoint matrix for V) as well as a matrix S that is a quadratic and a positively defined diagonal matrix or contains a quadratic and positively defined diagonal matrix as a sub-matrix and, moreover, has only zeroes as elements. The diagonal elements Skk (what are referred to as the singular values of DNL of this xe2x80x9cdiagonal matrixxe2x80x9d S are arranged according to size in conformity with S11xe2x89xa7S22xe2x89xa7 . . . xe2x89xa7SLLxe2x89xa70.
The number of diagonal elements different from 0 is equal to the rank of DNL. If the signal x(n) were to contain no noise, i.e. only p exponential oscillations, then exactly p( less than L) diagonal elements, what are referred to as the xe2x80x9cprincipal componentsxe2x80x9d (PC) would differ from 0; the others would be equal to 0. When, however, noise is present, then the remaining diagonal elements are in fact generally xe2x89xa00 but are small compared to the principal components. They are set approximately equal to 0 (PC method). An approximation S_of S is thus obtained.
With this approximation S_instead of S in the SVD decomposition (8), one then obtains, according to
Dxe2x80x94NL=U*Sxe2x80x94*VHxe2x80x83xe2x80x83(9)
a p-rank approximation Dxe2x80x94NL for the matrix DNL. With DNL instead of DNL, the FBLP coefficients a(k) are then calculated from the equation system
[a(L), a(Lxe2x88x921), . . . , a(0)]*Dxe2x80x94NL=0xe2x80x83xe2x80x83(7a)
and, thus, the above-described steps 2 and 3 (Prony) are implemented.
Since the rank of Dxe2x80x94NL is equal to p( less than L), the equation system (7a) is under-defined, i.e. its solution is initially not unambiguous. However, the solution with the least       Norma    ⁡          (              a        ⁡                  (          k          )                    )        =                    ∑                  k          =          0                L            ⁢                        "LeftBracketingBar"                      a            ⁢                          (              k              )                                "RightBracketingBar"                2            
is unambiguous given the subsidiary condition a(0)=1, and this minimum norm solution is generally established for an under-defined, linear equation system A*x=b by
x=AH*(A*AH)xe2x88x921*b
see, for example, the textbook of Marple, page 77. It has been empirically found that this minimum norm solution of the equation system (7a) in the second step leads to extremely good results for the sought frequencies.
Instead of being implemented with the date of matrix DNL, these calculations can also be implemented with the covariance matrix DNL, DNLH.
An advantageous version of the described PC method derives in the xe2x80x9cstate spacexe2x80x9dpresentation (B. D. Rao, K. S. Arun, xe2x80x9cModel Based Processing Of Signals: A State Space Approachxe2x80x9d, Proc. IEEE 80, 283-309 (1992)). The point of departure is the p-rank approximation Dxe2x80x94NL from (9), namely
Dxe2x80x94NL=U*Sxe2x80x94*VHxe2x80x83xe2x80x83(9a)
It can be shown that the product of the two first factors U* S_ can be factorized according to             U_      *      S_        =          ⌊                                    h                                                hF                                                              hF              2                                                            ⋮                                                              hF              L                                          ⌋        ,
whereby h is a line vector having the length p, and F is the xe2x80x9cstate transition matrixxe2x80x9d with p rows and p columns. The eigenvalues and eigenvectors of F supply the signal frequencies fk and the appertaining amplitudes ck without having to calculate the FBLP coefficients a(k).
One can proceed in the following way for calculating F: in the matrix, U* S_ with L+1 rows and p columns, the last row is omitted once (the result would read G1) and the first row (G2) is omitted on another occasion. G1* F=G2 then applies. Due to L greater than p, this equation system is over-defined for F and has the LS solution
F(G1H*G1)xe2x88x921*G1H*G2
The matrix xcex9=Qxe2x88x921*F*Q, formed with a matrix Q in whose columns the eigenvectors of F reside, is a diagonal matrix having the diagonal elements xcex9kk=exp(2xcfx80xc2x7ixc2x7fk). The fk are the desired frequencies. For defining the amplitudes ck, A=U* Sxe2x80x94* Q and B=Qxe2x88x921* VH are formed. The amplitudes ck then derives for each k by multiplication of the element of the first row of A residing in the kth column by the element of the first column of B residing in the kth row and division by the corresponding eigenvalue of F according to
ck=A1kxc2x7Bk1/exp(2xcfx80xc2x7ixc2x7fk)
An object of the present invention is to specify an improved method with which the distances and/or velocities of a plurality of objects can be simultaneously exactly measured.
According to the present invention, a method and apparatus is provided for distance and speed measurement. Signals that propagate in a waveform are emitted and, after reflection at targets, are registered as measuring signals so that the measured signals comprise a frequency modulation that is dependent on a value or values of a quantity or quantities to be measured. A frequency spectrum of the measured signals is analyzed on the basis of a previously defined plurality of measured values. The analyzing occurs by acquiring the plurality of measured values by sampling the measured signal at equidistant time intervals and storing the measured values in a predetermined sequence. Except for equivalent conversions, a data matrix wherein the measured values are chronologically arranged as a double Hankel matrix is diagonalized according to a SVD method. An approximation of the diagonalized data matrix is determined in that diagonal elements are set equal to zero beginning from a predetermined limit or from a limit calculated from a distribution of the diagonal elements. Frequencies are calculated from remaining diagonal elements.